Defining polynomial
|
$( x^{3} + 2 x + 68 )^{7} + 73$
|
Invariants
| Base field: | $\Q_{73}$ |
| Degree $d$: | $21$ |
| Ramification index $e$: | $7$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{73}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{73})$: | $C_3$ |
| This field is not Galois over $\Q_{73}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $389016 = (73^{ 3 } - 1)$ |
Intermediate fields
| 73.3.1.0a1.1, 73.1.7.6a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 73.3.1.0a1.1 $\cong \Q_{73}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 68 \)
|
| Relative Eisenstein polynomial: |
\( x^{7} + 73 \)
$\ \in\Q_{73}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^6 + 7 z^5 + 21 z^4 + 35 z^3 + 35 z^2 + 21 z + 7$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $42$ |
| Galois group: | $F_7$ (as 21T4) |
| Inertia group: | Intransitive group isomorphic to $C_7$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $7$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.8571428571428571$ |
| Galois splitting model: | not computed |